In the paper *Floer cohomology of Lagrangian intersections
and pseudo-holomorphic disks 2*, in the part of the preliminaries the author considers a Hamiltonian action of the isometry group $G$ of $\mathbb{C}\mathbb{P}^n$ on $\mathbb{C}\mathbb{P}^n$. The action is described in terms of the dual of the lie algebra and then he claims that for $\xi$ in the lie algebra of a maximal torus $f_{\xi}(x):=\langle \Phi(x),\xi\rangle = \frac{x^T\xi x}{2\pi i\|x\|^2}$.Then it's claimed that we can pick a $\xi$ such that the flow $\psi_t$ of the vector field generated by it is periodic with period one , and that $\psi_t(\mathbb{R}\mathbb{P}^n)\cap \mathbb{R}\mathbb{P}^n = \operatorname{Crit}(f_{\xi})=n+1$.

Now I have tried to see this why this is true but I couldn't. I am only used to working with hamiltonian actions of torus $\mathbb{T}^n$ , where there isn't any talk about lie algebras and their dual, so I was wondering if we could describe the action of $G$ as an action of a torus so that I could check the statements.

Or if anyone could enlighten me why this statements are true I would appreciated it.

Thanks in advance.